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Kinematic Models

In the late 19th and early 20th centuries, physical models were widely used by mathematics educators to depict a wide range of mathematical ideas in three dimensions. Most models were constructed in Europe, in particular Germany, for use in school and college instruction. These models could be static or articulated (movable). They were used to show mathematical surfaces and curves as well as other mathematical concepts useful to mathematicians, engineers and scientists. As more and more American institutions added mathematical models to their collections, American firms grew up to supply this growing demand. By World War I interest in these physical models waned, and most were relegated to closets and basements. But with the advent of computer graphics systems, the interest in physical mathematical models is returning.

The vast majority of mathematical models made around the turn of the 20th century were static (not able to be manipulated) and made of wood, paper or plaster. The paper models of American mathematician A. Harold Wheeler are a prime example. A common model still used in many high schools is the wooden or plaster cone used to show how the conic sections (circle, ellipse, parabola, hyperbola) arise from slicing a cone at different angles to its axis.

Less common are kinematic models. These devices depict mathematical concepts that involve motion. The models can be manipulated through the use of cranks and hinges and were usually constructed of metal, most often brass.

The most widely used kinematic model of the time was the linkage. A linkage is a device made of hinged armatures linked at pivot points to allow the whole assembly to move and deform in order to transform one type of movement into another. The most common and useful linkages are those that transform circular movement into linear movement, known as inversors. Circular motion is easy to produce using any sort of wheel device. However, true linear or straight line motion is not. To create linear motion, the circular motion of a crankshaft needs to be transformed into straight line motion. A common example of the conversion between linear and rotary motion is found in internal combustion engines. The piston’s linear motion is converted into the rotary motion of the crank shaft, which is in turn transmitted to the wheels. In 1784, Scottish engineer James Watt developed a system of rods and pins that allowed the pistons in his steam engines to exert force on both the downward and upward stroke of the piston (as opposed to in just one direction as in previous designs). This led to many advances in engineering. However, Watt’s design did not create exact linear motion. Along with straight line motion, linkages can generate oscillating motion from circular motion, again useful in engineering.

The second types of kinematic models are those that generate curves, which are used in diverse theoretical and engineering processes. Starting in the 17th century, one of the most studied types of curves has been the trochoids; curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve. Included in the family of trochoids are the cycloids (a circle rolling along a straight line), epitrochoids (a circle rolling around the outside of another circle) and the hypotrochoids (a circle rolling around the inside of another circle). The popular Spirograph drawing toy produces various trochiods.

The Spirograph toy produced by Hasbro was created by British engineerDenys Fisherand first sold in 1965 (object 2005.0055.02)

These types of curves have applications in engineering with respect to the workings of gears and motors, as well as answer interesting questions in mathematics, physics and engineering. In particular, gear teeth shaped using cycloidal and involute curves reduce friction and torque, allowing gears to rotate more efficiently.

During the second half of the 19th century, mathematical models for use in teaching became very popular. One of the most prominent producers (known as publishers) of models for commercial use was the firm of Ludwig Brill of Darmstadt, Germany. In 1899 Martin Schilling took over and expanded the business from Brill. The kinematic models in the Smithsonian’s collection housed at the National Museum of American History were produced by the firm of Martin Schilling. His firm produced numerous types of mathematical models, including twelve kinematic models of which the Smithsonian presently has ten. Originally located in Halle, Germany, the firm moved to Leipzig sometime after 1903. The kinematic models were designed by German mathematician Frederick Schilling (1868—1950), professor of mathematics at Göttingen, who became the scientific director of the company. The kinematic models are listed in the 1911 Schilling Catalog mathematischer Modelle, Leipzig. This catalog lists 377 items divided into forty series or types of models. Series XXIV consists of Kinematic models (Kinematische Modelle), models of movement. This series is divided into 4 groups:

Group 1 (models No 1-4): Trochiods (Epitrochoid and Hypotrochoid)

Group 2 (models No 5-7); Cyclic curves (including cycloids)

Group 3 (models No 8, 9): Twin cranks

Group 4 (models No 10-12): Inversors by Peaucellier, Hart, and Sylverster and Kempe

All of the Schilling models in the NMAH collection, with the exception of the Peaucellier Inversor, were given to the Smithsonian by the University of Michigan’s Department of Mathematics in 1964.The Peaucellier Inversor came to the museum from Brown University in 1973.

Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the tenth in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

Linkages are joined rods that move freely about pivot points. A pair of fireplace pincers is an example of a very simple linkage. Producing straight line motion was an important component of many machines. But producing true linear motion is very difficult. One area of research during the 19th century was to use linkages to produce linear motion from circular motion. In this context “inverse” is a geometric term that refers to the process of using algebra and trigonometry to convert or invert one geometric shape into another. In this case, the inverse of the circle will be a straight line. So an “inversor" is a device that finds the inverse of a geometrical object: the conversion of a circle to a straight line in the case of this model.

In 1864 French engineer Charles Nicolas Peaucellier (1832-1913) created a seven-bar linkage which succeeded in producing pure linear motion. Since then, such seven-bar linkages are often referred to as “Peaucellier cells” or a “Peaucellier’s inversor.” His discovery was the first solution of what was referred to as the problem of parallel motion: converting rotational to linear motion using only “rods, joints and pins.”

This linkage is constructed of seven metal armatures (two longer arms of 15 cm, four shorter arms of 5 cm) hinged so as to create a kite shape with a rhombus (diamond) shape at the top of the kite. This horizontal assembly is then attached to a central vertical axis that is rotated by turning a crank below the baseplate. Fingerholds are attached to the two primary hinge points to allow the linkage to be articulated from above. Below these fingerholds are metal points used to trace the motion of the linkage on the paper.

The bottom (tail end) of the kite is fixed on a circle. As the dial is turned or the fingerholds are moved, the point internal to the linkage traces the circle, resulting in the point at the top of the kite tracing a straight line. The seven-bar linkage works by keeping one end of the linkage (the tail end of the kite) fixed on a circle. As the center point traces around the circle, the point at the opposite end of the linkage (the top of the kite) traces a straight line. The German title for this model is: Inversor von Peaucellier 1864.

This model is marked Halle a.S., but this is lined through and has a Leipzig stamp. In or after 1903 Schilling moved the business from Halle to Leipzig. On top of the mounting plate is an aged paper sheet showing the name and maker of the linkage. Printed on the paper are a black circle and a red circle to show the circular path; a red curve; and, what was most likely a black linear path, but which is now worn through the paper with use.

Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the eleventh in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

Linkages are joined rods that move freely about pivot points. A pair of fireplace pincers is an example of a very simple linkage. Producing straight-line motion was an important component of many machines. But producing true linear motion is very difficult and one area of research during the 19th century was to use linkages to produce linear motion from circular motion. In this context, “inverse” is a geometric term that refers to the process of using algebra and trigonometry to convert or invert one geometric shape into another. In this case, the inverse of the circle will be a straight line. So an “inversor” is a device that finds the inverse of a geometrical object: the conversion of a circle to a straight line in the case of this model.

The Hart’s Inversor, also known as Hart’s Cell or Hart’s Linkage, is a contraparallelogram of four pin-connected links. It is similar to the Peaucellier Inversor, but is a four-bar linkage as opposed to a seven-bar linkage. It was invented and published by Harry Hart (1848-1920) in 1874.

This model is made from four metal armatures, two measuring 9.5 cm, two 16.5 cm, in an “hourglass” configuration (the two longer arms crossing to form the waist of the hourglass) with two congruent triangles meeting at a common vertex.

When the top and bottom arms are parallel to the top and bottom of the baseplate, the triangles are isosceles. The top arm is fixed to the base slightly to the right of its midpoint. Below this fixed point, a fifth arm is attached to a crank below the baseplate and attached to the underside of the upper cross arm slightly above the midpoint. This attachment can be rotated in a circle either by turning the crank or by using the polished fingerhold on the top of the cross arm.

A pin below the fingerhold (now inserted into a piece of cork to avoid tearing the paper covering of the baseplate) traces part of a circle as seen in the image. This causes a fingerhold and pin (also in a piece of cork) on the second cross arm, slightly below its midpoint, to move laterally right and left across the baseplate in a straight-line motion. As the attachment is rotated, the triangles become progressively more scalene.

In addition, this linkage has the following linearity property. When the linkage is in its original (isosceles) configuration, mark four points on each of the four arms such that the four points lie on a vertical line. Fix the top point and allow the second point (below the top point) to trace a circle. This causes the third point to trace a straight line, and all four points will remain colinear regardless of the configuration of the linkage.

The German title of this model is: Inversor von Hart. The name plate on the model gives a date of 1874 for this model, most likely indicating the date of Hart’s discovery.

Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the twelfth in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

Linkages are joined rods that move freely about pivot points. A pair of fireplace pincers is an example of a very simple linkage. Producing straight-line motion was an important component of many machines. But producing true linear motion is very difficult and one area of research during the 19th century was to use linkages to produce linear motion from circular motion. In this context, “Inverse” is a geometric term that refers to the process of using algebra and trigonometry to convert or invert one geometric shape into another. In this case, the inverse of the circle will be a straight line. So an “inversor” is a device that finds the inverse of a geometrical object: the conversion of a circle to a straight line in the case of this model.

A generalization of Hart’s Inversor, the Sylvester-Kempe Inversor is also known as a Quadruplane inversor and creates linear motion from circular motion. English mathematicians James Sylvester (1814-1897) and Alfred Kempe (1849-1922) developed the geometric theory behind these linkages in the 1870s. Kempe proved that every algebraic curve can be generated by a linkage using a Watt’s curve, after Scottish engineer James Watt (1736-1819).

Unlike the other Schilling linkages in the collection, this one is not made of armatures. It consists of linked triangular metal plates (two large and two small). The smallest triangle is attached to the baseplate at a stationary pivot point. The triangles are linked together at the vertices to form a chain of triangles (small-large-small-large). As with the other linkages, this model has an armature that is attached to a small hand crank on the underside of the baseplate and attached to the vertex of one of the larger triangles that allows the linkage to rotate. It can also be moved by using one of two fingerholds attached to the top of two of the triangles at a vertex.

As the linkage is rotated, a pin where the armature attaches to the large triangle traces out a circle, visible in the image. At the same time, a pin under the fingerhold on the opposite large triangle traces a straight line from left to right across the baseplate, also seen in the image. The German title of this model is: Inversor von Sylvester und Kempe. The nameplate on the model gives a date of 1875 for this model, most likely the date of discovery by Sylvester and Kempe.

Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the sixth in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

An involute of a circle is a curve that is produced by tracing the end of a string that is wrapped around a circle as it is unwound while being kept taut. It is the envelope of all points that are perpendicular to the tangents of a circle.

As with the three trochoidal models, these curves were used in the shaping of gear teeth in the 18th century. Following that, it was discovered that shaping the teeth of gears using the curve formed by the involute of a circle also increases the efficiency of gearage. Surprisingly, there are many applications of noncircular gears, such as elliptical, triangular, and quadrilateral gears. (See model 1982.0795.06.)

In this model a toothed circular gear of radius 13 mm is mounted on the baseplate and can be turned via a crank on the underside of the baseplate. A thick piece of beveled glass is mounted above the apparatus. A dark metal toothed bar 45 mm long is attached to the circular gear so that as the crank turns the circular gear, the toothed bar is forced past the circular gear and rotates round it.

Perpendicular to the bar is a thin clip with three small colored balls. A blue ball is attached at the edge of the bar where the bar will touch the circle and traces the involute of the circle in blue on the glass. A red ball is placed 33mm in front of the toothed side of the bar and produces a “stretched” involute in red. A green ball is 45mm behind the toothed side of the bar traces another “stretched” involute in green. German title is: Erzeugung von Kreisevolventen.

Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the seventh in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

The cycloid solves the 17th-century problem posed by Swiss mathematician Johann Bernoulli known as the brachistochrone problem. This problem asks for the shape of the curve of fastest decent: the path that a ball would travel the fastest along under the influence of gravity.

The cycloids are drawn by tracing the location of a point on the radius of a circle or its extension as the circle rolls along a straight line. Cycloids are members of the family of curves known as trochoids, curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve. The curve generated by a point on the circumference of the rolling circle is called an epicycloid, and a ball rolling on this curve (inverted) would travel faster than on any other path (the brachistochrone problem). Points either inside or outside the rolling circle generate curves called epitrochoids. The cycloid also solves the tautochrone problem, a curve for which a ball placed anywhere on the curve will reach the bottom under gravity in the same amount of time.

An example of the application of the cycloid as a solution of the tautochrone problem is the pendulum clock designed by Dutch physicist Christopher Huygens. As the width of the swing of the pendulum decreases over time due to friction and air resistance, the time of the swing remains constant. Also, cycloidal curves are used in the shaping of gear teeth to reduce torque and improve efficiency.

This model consists of a toothed metal disc linked to a bar that is toothed along one edge. A radius of the circle extending away from the bar has a place for a pin inside the circumference, a pin on the circumference, and a pin outside the circle. Rotating a crank below the baseplate of the model moves the circle along the edge of the bar, generating a curve above each point. The curves are indicated on the glass overlay of the mechanism. The curve generated by the point on the circumference of the circle is an epicycloid, depicted in blue on the glass; that generated by the point outside the circle is a prolate (from the Latin to elongate) cycloid, depicted in orange; and that generated by the point inside the circle is a curtate (from the Latin to shorten) cycloid, depicted in green. The German title of this model it: Erzeugung von Cycloiden (to produce cycloids).

Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the first in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

Circles rolling around the outside of other circles, known as epicycles to the ancient Greeks, were used to describe the motions of the planets in a geocentric cosmology. These curves,called epitrochoids, are formed by tracing a point on the radius or the extension of the radius of a circle as it rolls around the outside of a second stationary circle.

Epitrochoids are members of the family of curves called trochoids, curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve. They include the cycloids (see item 1982.0795.05) and hypotrochoids (see items 1982.0795.02 and 1982.0795.03). In the 18th century, it was found that when shaping the sides of gear teeth as the valley between teeth, using trochoidal curves reduced the torque of the rotating gears and allowed them to rotate more efficiently.

Depending on the distance of the tracing point from the center of the rolling circle, an infinite number of curves can be formed. The three curves depicted in this model are bicyclic, meaning the smaller circle needs to rotate around the larger circle twice before returning to is original configuration. The ratio of the radii of the two circles will determine the number of nodes in the curve and how many rotations are required before the tracing point returns to its starting configuration. The Spirograph toy produces various types of epitrochoids. (See item 2005.0055.02)

In this model, a toothed metal disc of radius 30 mm links to a smaller toothed metal disc of radius 12mm. Rotating a crank beneath the baseplate rolls the smaller disc around the outer edge of the larger disk. The model illustrates three curves that may be generated by the motion of a point at a fixed distance along the radius of a circle when the circle rolls around the outer edge of a larger circle.

The green point within the smaller circle (at radius 4mm) produces the green curve on the glass overlay of the model. The blue point on the circumference of the smaller circle (in this special case, the curve is known as an epicycloid) produces the blue curve. The third, represented by a red curve on the glass, is on the extension of a radius of the smaller circle (20mm). As the smaller circle rolls, the point moves inside the larger circle. The German title of this model is: Erzeugung der Epitrochoiden als solche mit freiem Centrum.

Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the third in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

The name hypotrochoid comes from the Greek word hypo, which means under, and the Latin word trochus, which means hoop. Thus hypotrochoids are curves formed by tracing a point on the radius or extension of the radius of a circle rolling around the inside of another stationary circle.

Hypotochoids are members of the family of curves called trochoids, curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve. They include the cycloids (see item 1982.0795.05) and epitrochoids (see item 1982.0795.01).

An infinite number of hypotrochoids can be formed, depending on the distance of the tracing point from the center of the rolling circle. The ratio of the radius of the rolling disc to the radius of the outer ring will determine the number of nodes the hypotochoid will have. In this model, the curves each have five nodes. Hypotochoids, for which the tracing point is on the extension of the radius, form curves that resemble petalled flowers and are called roses. The Spirograph toy produces various types of hypotrochoids. (See item 2005.0055.2) In the 18th century, it was found that shaping the sides of gear teeth and the valley between teeth by using trochoidal curves reduced the torque of the rotating gears and allowed them to rotate more efficiently.

This model consists of a stationary toothed metal ring (with teeth on the inner edge of the ring) of radius 80 mm. A toothed metal disc of radius 32 mm is attached to a brass arm of 7 cm that can be rotated by turning a crank below the baseplate. As the arm is rotated, the disc rolls around the inside of the ring. Three points lie along the radius of the disk and trace corresponding curves, or roulettes, on the glass overlay.

The blue point on the circumference of the disc traces a blue five-pointed star shape referred to as a hypocycloid. The green point on the radius of the disc traces a green curve inside the ring, and the red point on the extension of the radius of the disc traces a curve that extends past the radius of the ring. The German title of this model is: Erzeugung der Hypotrochoiden als soche mit freiem Centrum.

Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the fourth in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

The name hypotrochoid comes from the Greek word hypo, which means under, and the Latin word trochus, which means hoop. Thus hypotrochoids are curves formed by tracing a point on the radius or extension of the radius of a circle rolling around the inside of another stationary circle.

Hypotochoids are members of the family of curves called trochoids; curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve, and include the cycloids (see item 1982.0795.05) and epitrochoids (see item 1982.0795.01). An infinite number of hypotrochoids can be formed, depending on the distance of the tracing point from the center of the rolling circle. The ratio of the radius of the rolling disc to the radius of the outer ring will determine the number of nodes the hypotochoid will have.

In this model, the curves each have five nodes. Hypotochoids, for which the tracing point is on the extension of the radius, form curves that resemble petalled flowers and are called roses. The Spirograph toy produces various types of hypotrochoids. (See item 2005.0055.2) In the 18th century, it was found that shaping the sides of gear teeth, and the valley between teeth, using trochoidal curves reduced the torque of the rotating gears and allowed them to rotate more efficiently.

As with Schilling’s model number 3 of a hypotrochoid, this model has a stationary toothed metal ring of radius 60 mm. A toothed disc of radius 36 mm rolls around the inside of the ring by the use of a crank below the baseplate. However, this model has a semitransparent glass disc of the same radius as the ring attached to the rotating disc.

Traced on this glass disc is a red epitrochoid that would be formed by an imaginary point on the extension of the radius of a circle rotating on the outside of the disc in the model. A green point on this curve traces a green star-shaped hypotrochoid on the stationary glass overlay of the model as the disc is rotated.

The hypotrochoid can also be generated by imagining a point on the extension of the radius of the rotating disc. An orange point on the green hypotrochoid aligns with the epitrochoid and shows how the epitrochoid can be generated from a hypotrocoid and vice-versa. The German title of this model is: Erzeugung der Hypotrochoiden als soche mit bedecktem Centrum.

Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the eighth in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

Many machines need to produce a back and forth motion, such as the back and forth motion of the rods of a locomotive that drives the wheels. This back and forth motion is achieved by converting circular motion (produced by the pistons of the steam engine) to linear motion (of the rods). One way of achieving this in a smooth way is through a quick return mechanism. This model uses two ellipses that are held in constant contact, producing an “elliptical gear.”

As one ellipse rotates around the other, the distance between the fixed focus of one ellipse and the free focus of the other remains constant. This can be seen in the model by the placement of the arm. As the ellipses rotate about each other, the speed of rotation increase as the ellipses move towards a side-by-side orientation, and slows as the ellipses move towards an end-to-end alignment. Thus the velocity increases and decreases periodically as the ellipses rotate. The velocity ratio of the rotating gear is the portion of the length of the top arm over one ellipse divided by the remaining length (over the other ellipse.) Mathematically this velocity ratio varies from e/(1-e) to (1-e)/e where e is the eccentricity of the (congruent) ellipses. The cyclic nature of the velocity of this motion is known as a “quick-return” mechanism, which converts rotational motion into reciprocating or oscillating motion.

This model employs two identical elliptical metal plates (major axis 8 cm, minor axis 5 cm). Both ellipses were fixed to the baseplate at their right foci (though one ellipse is now detached) while the other foci are free. This allows the two ellipses to rotate around each other while remaining in contact. An 8 cm rigid arm connects the fixed foci of one ellipse to the free foci of the other.

Beneath the free foci of the left ellipse is a metal point. As the (now missing) crank below the baseplate is rotated, the point traces out a circle on the paper covering of the baseplate. Using the thumb hold at the midpoint of the arm, the two ellipses can be made to rotate around each other. A small ball-type joint at the ends of the major axis of each ellipse allows the two ellipses to join together when they are aligned end-to-end. The German title of the model is: Gleichläufiges Zwillingskubelgetriebe mit seinen Polbahnen (same shape transmitted by twin cranks with their poles).

Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the ninth in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

This model is an example of a Watt’s linkage. Linkages are joined rods that move freely about pivot points used to produce a certain type of motion. A pair of fireplace pincers is an example of a very simple linkage. Producing straight line motion was an important component of many machines. But producing true linear motion is very difficult and one area of research during the 19th century was to use linkages to produce linear motion from circular motion. Scottsman James Watt (1736-1819), devised linkages to create linear motion for use in early steam engines. A Watt’s linkage is a three-bar linkage in which two bars of equal length rotate to produce congruent circles. The ends of these two radii are joined by a longer crossbar. As the radii counter-rotate, the midpoint of the crossbar traces out a Watt’s Curve.

Watt’s Curve is related to the lemniscate, or a figure-eight-shaped curve. However, Watt’s Curve resembles a figure eight that has been compressed vertically so that the two lobes appear as circles that are flattened where they meet. As the midpoint of the crossbar traces the region of the lemniscate where the curve crosses itself, the motion is approximately linear.

This model consists of two identical components (“bowties”), each comprised of two rounded hyperbolic metal plates (13 cm base, 5.5 cm altitude) joined by an armature of 9 cm. One bowtie is mounted on top of and offset by 7 cm from the bottom bowtie. An armature attaches the vertex of one plate to the vertex of its corresponding plate below. A crank below the baseplate connects to one arm. When the crank is rotated, the two connecting arms rotate in opposing circular paths, causing the top bowties to follow a roughly figure eight path. As each arm rotates through 180 degrees, the bowties align first to the left, then to the right. The German title is of this model it: Gegenläufiges Zwillingskubelgetriebe mit seinen Polbahnen.